If rate of diffusion of A is 12 times that of B, what will be the density ratio of A and B?

Options

A. 144:1
B. 12:1
C. 1:12
D. 1:144

Explanation

To solve the problem of finding the density ratio of gases A and B given that the rate of diffusion of A is 12 times that of B, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The rate of diffusion of a gas is inversely proportional to the square root of its density. This can be expressed mathematically as:
\[ R \propto \frac{1}{\sqrt{D}} \]
where \( R \) is the rate of diffusion and \( D \) is the density of the gas.

2. **Setting Up the Equation**:
For gases A and B, we can express this relationship as:
\[ \frac{R_a}{R_b} = \sqrt{\frac{D_b}{D_a}} \]
where \( R_a \) and \( R_b \) are the rates of diffusion of gases A and B, respectively, and \( D_a \) and \( D_b \) are their densities.

3. **Substituting the Given Information**:
According to the problem, the rate of diffusion of A is 12 times that of B:
\[ R_a = 12 R_b \]
Substituting this into our equation gives:
\[ \frac{12 R_b}{R_b} = \sqrt{\frac{D_b}{D_a}} \]
Simplifying this, we find:
\[ 12 = \sqrt{\frac{D_b}{D_a}} \]

4. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ 12^2 = \frac{D_b}{D_a} \]
This simplifies to:
\[ 144 = \frac{D_b}{D_a} \]

5. **Finding the Density Ratio**:
To find the ratio of the densities \( \frac{D_a}{D_b} \), we take the reciprocal of the above equation:
\[ \frac{D_a}{D_b} = \frac{1}{144} \]
This can also be expressed as:
\[ D_a : D_b = 1 : 144 \]

### Final Answer:
The density ratio of A to B is \( 1 : 144 \).